Question

Find the Laplace transform of each of the following functions: a) \(f(t)=-20 e^{-5(t-2)} u(t-2)\) b) \(f(t)=(8 t-8)[u(t-1)-u(t-2)]\) \(+(24-8 t)[u(t-2)-u(t-4)]\) \(+(8 t-40)[u(t-4)-u(t-5)]\)

Short answer

(a) \(\frac{-20 e^{-2s}}{s+5}\); (b) Combine: \(\frac{8 e^{-s}}{s^2} + (\frac{24}{s} - \frac{8}{s^2}) e^{-2s} + (\frac{8}{s^2} - \frac{40}{s}) e^{-4s}\)."

Step-by-step solution

Understanding the Laplace Transform

The Laplace Transform of a function \( f(t) \) is given by \( \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt \). It is particularly useful for transforming piecewise and discontinuous functions into the s-domain, making differentiation and integration easier.

Expressing the Function in Standard Form (Part a)

Given \( f(t) = -20 e^{-5(t-2)} u(t-2) \), substitute \( t' = t-2 \) resulting in \( f(t') = -20 e^{-5t'} \), which is a function of \( t' \). The translation by 2 allows using the shifting property in Laplace transforms.

Shifting Property Application (Part a)

The shifting property states \( \mathcal{L}\{f(t-a) u(t-a)\} = e^{-as} F(s) \), where \( F(s) \) is the Laplace transform of \( f(t) \). For \( f(t) = -20 e^{-5t} \), the transform is \( \frac{-20}{s+5} \). Applying the property, \( \mathcal{L}\{f(t)\} = e^{-2s} \frac{-20}{s+5} \).

Simplifying the Laplace Transform (Part a)

Substitute back to obtain \( \mathcal{L}\{f(t)\} = \frac{-20 e^{-2s}}{s+5} \). This is the Laplace transform for part (a).

Analyzing the Function for Part (b)

For part (b), we have a piecewise function expressed in terms of unit step functions: \((8 t - 8)[u(t-1) - u(t-2)] + (24 - 8 t)[u(t-2) - u(t-4)] + (8 t - 40)[u(t-4) - u(t-5)]\). Each segment can be considered separately for simplification and transformed using the step function properties.

Solving the Laplace Transform for Each Segment (Part b)

Using the properties of Laplace transforms and linearity, address each function segment separately:- Segment 1: \((8 t - 8)[u(t-1) - u(t-2)]\) 1. Shift by 1 to get \(g(t) = 8(t') + 0u(t')\), transform to \(\frac{8}{s^2}\). 2. Use shift: \(\mathcal{L} = \frac{8 e^{-s}}{s^2}\).- Segment 2: \((24 - 8 t)[u(t-2) - u(t-4)]\) 1. Shift by 2 to transform \(g(t) = 24 - 8 t'\). 2. Transform: \(\mathcal{L} = \frac{24}{s} - \frac{8}{s^2}\). 3. Use shift: \(\mathcal{L} = \left( \frac{24}{s} - \frac{8}{s^2} \right) e^{-2s}\).- Segment 3: \((8 t - 40)[u(t-4) - u(t-5)]\) 1. Shift by 4 to transform \(g(t) = 8(t') - 40\). 2. Transform: \(\mathcal{L} = \frac{8}{s^2} - \frac{40}{s}\). 3. Use shift: \(\mathcal{L} = \left( \frac{8}{s^2} - \frac{40}{s} \right) e^{-4s}\).

Combining Transforms for Part (b)

For part (b), add results of all segments: \[ \mathcal{L}(f(t)) = \frac{8 e^{-s}}{s^2} + \left( \frac{24}{s} - \frac{8}{s^2} \right) e^{-2s} + \left( \frac{8}{s^2} - \frac{40}{s} \right) e^{-4s} \].This accounts for each discontinuity in the piecewise function.

Key concepts

Piecewise Functions

Piecewise functions are mathematical expressions that are defined by different sub-functions over separate intervals of the independent variable. These functions enable more flexible modeling of complex, real-world scenarios by allowing different behavior in distinct segments of the domain. Typically, piecewise functions are represented as a series of conditions or intervals, each associated with its own function equation.

• For example, a piecewise function might be defined such that for time periods 0 to 10 seconds, function A applies, while for time periods 10 to 20 seconds, function B applies.
• This segmented approach simplifies the handling and transformation of functions, especially when dealing with non-continuous scenarios.
• In practice, piecewise functions often come alongside unit step functions, which define clear boundaries between the segments.

In the context of Laplace transforms, piecewise functions can be transformed into the s-domain to facilitate analysis and solve complex problems, particularly those involving differential equations.

Shifting Property

The shifting property in the realm of Laplace transforms is an essential concept that allows transformation of functions that are time-shifted. Essentially, this property simplifies the Laplace transformation by addressing the time-shifting aspect separately.

The formula for the shifting property is given by:

• If you have a function expressed as \(f(t-a)u(t-a)\), its Laplace transform is \(e^{-as}F(s)\).
• Here, \(F(s)\) represents the Laplace transform of the function without the time-shift.

This property is particularly useful when dealing with delayed or retarded functions, common in control systems and signal processing. The time-shift essentially introduces an exponential multiplicative factor, \(e^{-as}\), corresponding to the delay of \(a\) in the time domain.

By utilizing the shifting property, one can systematically calculate the Laplace transforms of time-shifted functions by applying the shift separately, simplifying the overall process.

Unit Step Functions

Unit step functions, often known as Heaviside functions, are critical in piecewise function analyses because they essentially dictate when portions of a piecewise function become active or inactive.

These functions are denoted as \(u(t-a)\), where they turn 'on' at time \(t = a\), shifting from 0 to 1.

• The use of unit step functions simplifies expression of sections by distinctly marking each part of piecewise functions.
• They act like switches, ensuring function pieces are accounted for only within their intended time frames.

In complex functions like those addressed in Laplace transform applications, unit step functions mark the transition of segments and are crucial in defining the effective domain of each piece. This makes them indispensable in control engineering and systems analysis whereby timing and event-triggered actions are commonplace.

Essentially, they help manage and analyze transition points clearly, facilitating smooth integration into the s-domain.

S-Domain

The s-domain is a frequency domain that results from the Laplace transform of a time domain function. Transitioning functions from the time domain to the s-domain offers a formatted way of examining their behavior, especially in systems analysis and engineering challenges.

Within the s-domain:

• Functions are expressed in terms of \(s\), a complex frequency variable, simplifying differential equations into algebraic equations.
• This simplification makes solving complex circuit and dynamic system problems more manageable.
• It provides a means to analyze system stability and behavior, useful in control theory and signal processing.

By leveraging the s-domain, insights into the system's response, stability, and transient states—profound aspects of both theoretical and applied physics—become readily discernible.

The manipulation of functions within this domain is straightforward, with many mathematical rules, such as partial fractions and convolution, readily applicable to aid in inverse transformation to the time domain when required.